VN:F [1.9.22_1171]please wait...Rating: 5.0/5 (1 vote cast)How to find all possible triples from only one side of right angle triangle LOGICAL EXPLANATION My formula is based upon the following equality Hence if D is chosen in such a manner that it is a factor of x2 then we will always have integer values for Pythagorean Triples since D will get canceled from numerator & denominator Moreover D should be such that X2 / D is ‘Odd’ if D is Odd or it is ‘even’ if D is ‘even’ since even +/- even = even & odd +/- odd =even, denominator’s 2 will also get canceled. We should note that D can also be greater than X, since we get negative value in such cases we should take absolute figure. Thus if, First side of triple = X then, Second Side = |(X2 – D2)/2D| Hypotenuses = (X2 + D2)/2D For defining solution set for D, we first have to express X in terms of 2n * y. (where y is an odd number) Let set ‘A’ contain factors of y2 Solution set for D = {21 * all members of set A U 22 * all member of set A U … 2n * all member of set A} (Here ‘U’ stands for union) [Note:- While solving last part (i.e. 2n * all member of set A), one should stop as soon as 2n * any member of A = X and remaining members of A need not be consider since they will give same answer as first half element of that particular subset.] Examples: 1) if X = 35 then it means X = 20 * 35 hence y = 35. Hence set A will contain factors of 352 i.e. {1,5,7,25,35,49,175,245,1225} However Solution set for D will be {1,5,7,25} [as 20*5th element of set = 35 we need to consider only till 4th element]. Hence there will be 4 triples having 35 as one of the sides. Triples in this case will be (35,612,613),(35,120,125),(35,84,91),(35,12,37) 2) if X = 24 then it means X = 23 * 3 hence y = 3....

VN:F [1.9.22_1171]please wait...Rating: 4.5/5 (2 votes cast)A Brain Teaser which is very popular on net goes as below: 1. pick a number between 2 and 9 2. multiply it by 9 3. add the two digits in the new number not counting their place values 4. minus 5 from that answer 5. every number stands for a letter 1 – A, 2 – B, 3 – C, find your letter 6. think of a country that starts with that letter 7. take the last letter of that country and think of an animal 8. take the last letter of that animal and think of a fruit Now let’s analyze the problem step by step: First part: Guess a number and Multiply it with 9, then add its digits and reduce 5 from it. This part is just a simple distraction to make you feel you are doing something complex. As many of you may know multiplying any number by 9 and adding its digits till you get a single digit will always result in 9. (visit http://mathforum.org/library/drmath/view/67061.html for more information) Now when you remove 5 from 9, you are always left with 4 and Fourth letter of alphabet is “D”, As you may know: (1) You have four countries starting with “D” (a) Denmark (b) Djibouti (c) Dominica (d) Dominican Republic Hence Mathematically you will agree that you have 1/4 chances of selecting Denmark (2) Now If you have selected Denmark then what is the chance that you will select Kangaroo in the next round? Well you have following choices (a) Kangaroo (b) Kaffir cat (c) Kafue flats lechwe (d) Kelp gull (e) Killer whale (f) Kinkajou (g) Kirk’s dik dik (h) Klipspringer (i) Koala (j) Komodo dragon (k) Kongoni (l) Kudu So you have...